- #1
BlackWyvern
- 104
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I think I'm right when I say that the centroid of a 2D shape is found by the intersection of the lines that separate that shape into two shapes of equal areas.
Is that correct? I don't want to (for now) think about it in terms of moments and integrals, because frankly, it's a little confusing.
It would make sense if that's the case.
Then lastly, say we have a parabola, just a standard y = x^2. It's centroid will be found on the y axis, but the exact value is only able to be determined by calculus. I think I'm correct when I say that a section of parabola (made with a horizontal cut) will be similar to the parabola before the cut. Also as a result of the similarity, a parabola will take up the same amount of space for a given rectangle with vertexes on (0, 0) and (x, y).
Using this definition, we can say that the centroid of a parabola that extends to x = 10, y = 100 is found by this method:
x = 10
y = x^2 = 100
A_{rectangle}= xy = x^3 = 1000
A_{underparabola} = \int_{0}^{10} x^2 dx = 333.3333...
A_{rectangle} - A_{underparabola} = A_{parabola}
1000 - 333.333... = 666.666...
A_{parabola} / A_{rectangle} = P:A = 0.666...
P:A is the ratio this parabola takes of it's envelope rectangle (should be constant for all values of \infty > x > 0
Now the area of the parabola is halved (which gives the area of the lower, parabola shaped section:
666.666... / 2 = 333.333...
333.333... / P:A = A_{smallrectangle} = 500
500 = xy = x^3
x = \sqrt[3]{500}
y = x^2
y = 500^{2/3}
y ~ 63
Centroid = (0, 63)
I'm pretty sure this is correct, but can someone who's a bit more senior confirm for me?
Thanks.
Is that correct? I don't want to (for now) think about it in terms of moments and integrals, because frankly, it's a little confusing.
It would make sense if that's the case.
Then lastly, say we have a parabola, just a standard y = x^2. It's centroid will be found on the y axis, but the exact value is only able to be determined by calculus. I think I'm correct when I say that a section of parabola (made with a horizontal cut) will be similar to the parabola before the cut. Also as a result of the similarity, a parabola will take up the same amount of space for a given rectangle with vertexes on (0, 0) and (x, y).
Using this definition, we can say that the centroid of a parabola that extends to x = 10, y = 100 is found by this method:
x = 10
y = x^2 = 100
A_{rectangle}= xy = x^3 = 1000
A_{underparabola} = \int_{0}^{10} x^2 dx = 333.3333...
A_{rectangle} - A_{underparabola} = A_{parabola}
1000 - 333.333... = 666.666...
A_{parabola} / A_{rectangle} = P:A = 0.666...
P:A is the ratio this parabola takes of it's envelope rectangle (should be constant for all values of \infty > x > 0
Now the area of the parabola is halved (which gives the area of the lower, parabola shaped section:
666.666... / 2 = 333.333...
333.333... / P:A = A_{smallrectangle} = 500
500 = xy = x^3
x = \sqrt[3]{500}
y = x^2
y = 500^{2/3}
y ~ 63
Centroid = (0, 63)
I'm pretty sure this is correct, but can someone who's a bit more senior confirm for me?
Thanks.